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A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Question
Please answer correctly. Deals with permutations and combinations
![**Question:**
How many numbers between 1 and 100 (inclusive) are divisible by 5 or 3?
**Answer Field:**
(An empty space is provided for the user to input their answer.)
---
**Explanation:**
To solve this problem, the goal is to count all numbers in the specified range that are divisible by either 5 or 3.
1. **Divisibility by 5:**
- Numbers divisible by 5 between 1 and 100 are 5, 10, 15, ..., 100.
- These numbers form an arithmetic sequence with the first term 5, last term 100, and common difference 5.
2. **Divisibility by 3:**
- Numbers divisible by 3 between 1 and 100 are 3, 6, 9, ..., 99.
- These numbers form an arithmetic sequence with the first term 3, last term 99, and common difference 3.
3. **Divisibility by both 5 and 3 (i.e., 15):**
- Numbers divisible by 15 between 1 and 100 are 15, 30, 45, ..., 90.
- These numbers form an arithmetic sequence with the first term 15, last term 90, and common difference 15.
By applying the principle of inclusion-exclusion:
- Count numbers divisible by 5 [\( n_5 \)]
- Count numbers divisible by 3 [\( n_3 \)]
- Subtract numbers divisible by both to avoid double counting [\( n_{15} \)]
Total numbers = \( n_5 + n_3 - n_{15} \)](https://content.bartleby.com/qna-images/question/5e80208e-8c8d-42c4-a70b-dfa6cab9ca4b/06ef2f95-2712-4fee-bf62-783b5701f56d/pdwt0ar_thumbnail.png)
Transcribed Image Text:**Question:**
How many numbers between 1 and 100 (inclusive) are divisible by 5 or 3?
**Answer Field:**
(An empty space is provided for the user to input their answer.)
---
**Explanation:**
To solve this problem, the goal is to count all numbers in the specified range that are divisible by either 5 or 3.
1. **Divisibility by 5:**
- Numbers divisible by 5 between 1 and 100 are 5, 10, 15, ..., 100.
- These numbers form an arithmetic sequence with the first term 5, last term 100, and common difference 5.
2. **Divisibility by 3:**
- Numbers divisible by 3 between 1 and 100 are 3, 6, 9, ..., 99.
- These numbers form an arithmetic sequence with the first term 3, last term 99, and common difference 3.
3. **Divisibility by both 5 and 3 (i.e., 15):**
- Numbers divisible by 15 between 1 and 100 are 15, 30, 45, ..., 90.
- These numbers form an arithmetic sequence with the first term 15, last term 90, and common difference 15.
By applying the principle of inclusion-exclusion:
- Count numbers divisible by 5 [\( n_5 \)]
- Count numbers divisible by 3 [\( n_3 \)]
- Subtract numbers divisible by both to avoid double counting [\( n_{15} \)]
Total numbers = \( n_5 + n_3 - n_{15} \)
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